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Outline. Gear Theory Fundamental Law of Gearing Involute profile Nomenclature Gear Trains Loading Stresses. the angular velocity ratio of the gears of a gearset must remain constant throughout the mesh. gear. pinion. pitch circle, pitch diameter d, pitch point.

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outline
Outline
  • Gear Theory
    • Fundamental Law of Gearing
    • Involute profile
  • Nomenclature
  • Gear Trains
  • Loading
  • Stresses
fundamental law of gearing

the angular velocity ratio of the gears of a gearset must remain constant throughout the mesh

gear

pinion

pitch circle, pitch diameter d, pitch point

Fundamental Law of Gearing

functionally, a gearset is a device to exchange torque for velocity

P=T

What gear tooth shape can do this?

click here!

towards the involute profile
Towards the Involute Profile

A belt connecting the two cylinders

line of action, pressure angle 

profile of the involute profile
Profile of the Involute Profile

pressure angle, line of action, length of action, addendum

involute in action
Involute in Action

video from http://www.howstuffworks.com

nomenclature
Nomenclature

Figure 11-8

pitches etc
Pitches, Etc.

circular pitch (mm, in.)

base pitch (mm, in.)

diametral pitch (teeth/in.)

module (mm/teeth)

velocity ratio
Velocity Ratio

pitches must be equal for mating gears, therefore

contact ratio
Contact Ratio

average number of teeth in contact at any one time

=

length of action divided by the base pitch, or,

where C=center distance=(Ng+Np)*1/pd*1/2

minimum of teeth
Minimum # of Teeth

minimum # of teeth to avoid undercutting with gear and rack

outline12
Outline
  • Gear Theory
    • Fundamental Law of Gearing
    • Involute profile
  • Nomenclature
  • Gear Trains
  • Loading
  • Stresses
simple gear train16
Simple Gear Train
  • Fine for transmitting torque between
    • shafts in close proximity
    • when mv does not need to be too large
  • Use third gear (“idler”) only for directional reasons (not for gear reduction)
outline19
Outline
  • Gear Theory
    • Fundamental Law of Gearing
    • Involute profile
  • Nomenclature
  • Gear Trains
  • Loading
  • Stresses
loading of gears
Loading of Gears

Wt= tangential (transmitted) load

Wr= radial load

W= total load

Because Tp is constant, should Wt be static for a tooth?

(see board for figure or page 711 in book)

loading example
Loading Example
  • Given:
  • pinion is driving gear with 50hp at 1500 rpm
  • Np=20
  • mv=2 (a.k.a., mg=2)
  • pd=4 /in.
  • =20 degrees
  • Find:
  • Loading and effects of inputs on loading
outline23
Outline
  • Gear Theory
    • Fundamental Law of Gearing
    • Involute profile
  • Nomenclature
  • Gear Trains
  • Loading
  • Stresses
gear failure

Lewis, 1892, first formulation of gear tooth fatigue failure

Gear Failure

Fatigue Loading

Surface Failure

  • from bending of teeth
  • infinite life possible
  • failure can be sudden
  • from contact b/n of teeth
  • infinite life not possible
  • failure is gradual
agma gear stress formula

J Kv Km Ka Ks KB KI

AGMA Gear Stress Formula

many assumptions: see page 714…

…including that the contact ratio 1<mp<2

bending strength geometry factor

JKv Km Ka Ks KB KI

Bending Strength Geometry Factor
  • from tables on pgs. 716-718
  • Inputs
    • pinion or gear
    • number of teeth
    • pressure angle
    • long-addendum or full-depth
    • tip loading or HPSTC
finding j for the example

Jp=0.34, Jg=0.38

Finding J for the Example
  • pinion is driving gear with 50hp at 1500 rpm
  • Np=20, Ng=40
  • pd=4 /in.
  • =20 degrees
  • Wt=420 lb.
dynamic velocity factor

JKv Km Ka Ks KB KI

Dynamic (Velocity) Factor

to account for tooth-tooth impacts and resulting vibration loads

from Figure 11-22 or from Equations 11.16-11.18 (pages 718-719)

  • Inputs needed:
  • Quality Index (Table 11-7)
  • Pitch-line Velocity
      • Vt = (radius)(angular speed in radians)
determining k v for the example
Determining Kv for the Example

Vt=(1500rev/min)(2.5 in)(1ft/12in)(2 rad/1rev)=1964 ft/min

(well below Vt,max)

therefore, Qv=10

load distribution factor k m

JKvKm Ka Ks KB KI

Load Distribution Factor Km

to account for distribution of load across face

8/pd < F < 16/pd

can use 12/pd as a starting point

k m for the example
Km for the Example

F=12/pd=3 in.

Km=1.63

application factor k a

JKv KmKaKs KB KI

Application Factor, Ka

to account for non-uniform transmitted loads

for example, assume that all is uniform

other factors

JKv Km Ka Ks KB KI

Other Factors

Ks

to account for size

Ks=1 unless teeth are very large

KB

to account for gear with a rim and spokes

KB=1 for solid gears

KI

to account for extra loading on idler

KI=1 for non-idlers, KI=1.42 for idler gears

compared to what

similar concept to Se, but particularized to gears

Table 11-20

Figure 11-25 for steels

Sfb´

Sfb´ 

Hardness

Compared to What??

b is great, but what do I compare it to?

life factor k l
Life Factor KL

to adjust test data from 1E7 to any number of cycles

temperature reliability factors
Temperature & Reliability Factors

KT=1 if T < 250°F

see Equation 11.24a if T > 250°F

fatigue bending strength for example

Sfb´= 13ksi

Table 11-20

N=(450 rev/min)(60 min/hr)(8 hr/dy)(250 dy/yr)(10 yrs)=5.4E8 revs

KL=1.3558N–0.0178=0.948

Fatigue Bending Strength for Example

Class 40 Cast Iron

450 rpm, 8 hours/day, 10 years

T = 80°F

Reliability=99%

Sfb = (0.948)13ksi = 12.3 ksi

safety factor
Safety Factor

Nbpinion =(12300 psi)/(3177psi)= 3.88

Nbgear =(12300 psi)/(2842psi)= 4.33

gear failure42

Buckingham pioneered this work

equal to 1

same as for bending

d= pitch diameter of smaller gear

Gear Failure

Fatigue Loading

Surface Failure

  • from bending of teeth
  • infinite life possible
  • failure can be sudden
  • from contact b/n of teeth
  • infinite life not possible
  • failure is gradual
geometry factor i
Geometry Factor I

considers radius of curvature of teeth and pressure angle

top sign is for external gears, bottom for internal

For example: p=0.691, g=1.87, I=0.095

elastic coefficient
Elastic Coefficient

considers differences in materials

For example, Cp=1960 psi

surface fatigue strengths

from Table 11-21

same as for bending

Surface-Fatigue Strengths

c is great, but what do I compare it to?

life factor c l
Life Factor CL

to adjust test data fro 1E7 to any number of cycles

hardness ratio c h
Hardness Ratio CH

to account for pitting resistance

when pinion is harder than gear, then gear is cold-worked

only apply CH to the cold-worked gear

when HBp=HBG, then CH=1

safety factor for loading
Safety Factor for Loading

stress is based on the square root of loading, therefore:

gear design50

gear

pinion

Gear Design

Same for Pinion and Gear

  • pd (pc), , F
  • Power
  • W, Wt, Wr
  • Vt
  • Nc

Different for Pinion and Gear

  • d, N
  • T,  ,
  • Nb
gear design strategy
Gear Design Strategy

Given or Set:

gear ratio

Power, , T

Properties of Gears to Determine:

dp, dg, Np, Ng

pd, F -OR- Safety Factors

(the others are given)

Wt

Bending & Surface Stresses

Safety Factors or pd and F